Effective density of values of indefinite ternary inhomogeneous quadratic forms

TL;DR

This paper establishes effective lower bounds on the density of integer points near values of inhomogeneous indefinite ternary quadratic forms, with bounds depending explicitly on the Diophantine properties of the shift vector.

Contribution

It provides the first effective bounds for the distribution of values of inhomogeneous indefinite ternary quadratic forms, explicitly relating bounds to Diophantine properties of the shift.

Findings

Effective lower bounds for integer vectors near quadratic form values.

Bounds depend explicitly on Diophantine properties of the shift.

Applicable to algebraic shift vectors with bounds up to rac{1}{8}."

Abstract

Given an inhomogeneous quadratic form $Q_\xi(v)=Q(v+\xi)$ with $Q$ an indefinite $\mathbb{Q}$-isotropic rational ternary form and $\xi\in \mathbb{R}^3$ irrational, we prove an effective lower bound for the number of integer vectors $v\in \mathbb{Z}^n$ with $\|v\| \leq T$ such that $|Q_\xi(v)-t|<\delta$ that is valid for any $t\in \mathbb{R}$ and all $\delta\geq T^{-\nu}$, with $\nu>0$ depending explicitly on the Diophantine properties of $\xi$. In particular, for $\xi$ with algebraic entries we can take any $\nu<\frac{1}{8}$.